Let $p \in ]0,1[$ and $(X_n)_{n\geq0}$ be a Markov chain on E:={a,b,c} with transition matrix
$Q:=\begin{pmatrix} 1-p & p & 0 \\ 1/2 & 0 & 1/2 \\ 0 & 0 & 1 \\ \end{pmatrix}$
I should compute $\mathbb{P}(X_n=b|X_0=a)$ for $n \in \mathbb{N}$ and further the limit of it for $n \to \infty$.
I tried to use that $Q^n(a,b)=(Q^{n-1}\cdot Q)(a,b)$.
So I have: $\mathbb{P}(X_n=b|X_0=a)=(1-p)\mathbb{P}(X_{n-1}=b|X_0=a)+p\cdot \mathbb{P}(X_{n-1}=b|X_0=b)$ as I can only get from a to b...
I defined $a_{n}=\mathbb{P}(X_n=b|X_0=a)$ and $a_{n-1}=\mathbb{P}(X_{n-1}=b|X_0=a)$ and $b_{n-1}=\mathbb{P}(X_{n-1}=b|X_0=b)$. Further I have the initial values $a_0=1$ and $a_1=p$. If I'm in b I have to go back to a to able to go to b again... so $ b_{n-1}=1/2 \cdot a_{n-2}$.
Therefore I have the equation $a_n=(1-p) \cdot a_{n-1}+p/2 \cdot a_{n-2}$.
Unfortunately I don't know how to solve this equation using the initial condition. Am I on the right track? And what condition would I have to use to continue here?